Calculus Examples

$\sum_{x = 1}^{4} x^{3} + 3 x^{2} + 3 x + 1$

Step 1

Expand the series for each value of $x$ .

$1^{3} + 3 \cdot 1^{2} + 3 \cdot 1 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2

Simplify.

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Step 2.1

Raise $1$ to the power of $3$ .

$1 + 3 \cdot 1^{2} + 3 \cdot 1 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.2

Raise $1$ to the power of $2$ .

$1 + 3 \cdot 1 + 3 \cdot 1 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.3

Multiply $3$ by $1$ .

$1 + 3 + 3 \cdot 1 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.4

Add $1$ and $3$ .

$4 + 3 \cdot 1 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.5

Multiply $3$ by $1$ .

$4 + 3 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.6

Add $4$ and $3$ .

$7 + 1 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.7

Add $7$ and $1$ .

$8 + 2^{3} + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.8

Raise $2$ to the power of $3$ .

$8 + 8 + 3 \cdot 2^{2} + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.9

Raise $2$ to the power of $2$ .

$8 + 8 + 3 \cdot 4 + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.10

Multiply $3$ by $4$ .

$8 + 8 + 12 + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.11

Add $8$ and $12$ .

$8 + 20 + 3 \cdot 2 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.12

Multiply $3$ by $2$ .

$8 + 20 + 6 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.13

Add $20$ and $6$ .

$8 + 26 + 1 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.14

Add $26$ and $1$ .

$8 + 27 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.15

Add $8$ and $27$ .

$35 + 3^{3} + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.16

Raise $3$ to the power of $3$ .

$35 + 27 + 3 \cdot 3^{2} + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.17

Raise $3$ to the power of $2$ .

$35 + 27 + 3 \cdot 9 + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.18

Multiply $3$ by $9$ .

$35 + 27 + 27 + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.19

Add $27$ and $27$ .

$35 + 54 + 3 \cdot 3 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.20

Multiply $3$ by $3$ .

$35 + 54 + 9 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.21

Add $54$ and $9$ .

$35 + 63 + 1 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.22

Add $63$ and $1$ .

$35 + 64 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.23

Add $35$ and $64$ .

$99 + 4^{3} + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.24

Raise $4$ to the power of $3$ .

$99 + 64 + 3 \cdot 4^{2} + 3 \cdot 4 + 1$

Step 2.25

Raise $4$ to the power of $2$ .

$99 + 64 + 3 \cdot 16 + 3 \cdot 4 + 1$

Step 2.26

Multiply $3$ by $16$ .

$99 + 64 + 48 + 3 \cdot 4 + 1$

Step 2.27

Add $64$ and $48$ .

$99 + 112 + 3 \cdot 4 + 1$

Step 2.28

Multiply $3$ by $4$ .

$99 + 112 + 12 + 1$

Step 2.29

Add $112$ and $12$ .

$99 + 124 + 1$

Step 2.30

Add $124$ and $1$ .

$99 + 125$

Step 2.31

Add $99$ and $125$ .

$224$